There are many ways to deflect an asteroid. It depends on its size, and its orbit.

The main ones are

• With ten years, then to change it from a hit to a miss, you only need to deflect it by 6,371 km, and that’s the worst case where it’s a bullseye hit dead center. You don’t really change its direction, not by much anyway - rather you slow it down or speed it up so that it gets to the “rendezvous” a bit earlier or later, but the calculation is the same.
  
  Divide that by the number of seconds in ten years (10*365*24*60*60 ) and you get 6371000/ 315360000 = 0.02 meters per second, or about 2 centimeters per second, or about 22 hours per mile (0.045 miles per hour). That’s slightly faster than the average speed for a garden snail. It will be less than that if you need to prevent a glancing hit.
  
   Depending on its size then you may be able to do that with just a kinetic impact by another spacecraft hitting it at several kilometers per second (not hard to achieve). For instance a one ton mass hitting an asteroid at, say, 1 kilometer per second, would deflect a 50,000 ton asteroid by 2 centimeters per second.
  
  In the planetary defense conference early in (2015), delegates practiced a scenario with a hypothetical incoming 365 meter meteorite, how it would play out, with kinetic interceptor spacecraft. With the meteorite discovered in 2016, in this imaginary scenario, they send six kinetic impactor spacecraft, two fail, but the remaining four are able to deflect it so that most of it missed Earth in 2022, however one small fragment enters the atmosphere over Bangladesh but should break up during re-entry.
  
  Note - this is not a real event, it’s an imagined scenario though at the time it triggered some scare stories by people who took their imagined “press releases” to be real despite being clearly labelled as fictional. But it gives an idea of how a real world scenario like this might play out.
• If it does a flyby of Earth first, then it is likely to be very sensitive to small changes in velocity, as it has to go through a small “keyhole” of the order of hundreds of meters in diameter, to hit Earth next time around. So now you are talking about a delta v measured in microns per second. In this case, it might be enough to just paint it white and use the Yarkovsky effect
• There are so many asteroids, that you may be able to deflect a smaller asteroid to hit your larger asteroid. A small impact on the smaller asteroid could deflect it enough to make a much larger effect on the larger asteroid.
• You can move the asteroid by many other methods including the “gravity tractor”, ion beam shephard, you might try nuclear impact for more effect, you might use solar powered infrared lasers

Luckily there is no realistic chance of a really large asteroid hitting Earth. You can tell by the cratering record, because there are no impact craters that big for over 3 billion years anywhere in the inner solar system, on Mars, the Moon, moons of Mars, Earth, Mercury and what we have of the history of Venus. I’ll add this to my answer, just forgot to say it. The reason seems to be that Jupiter protects us from large objects beyond Jupiter, and the asteroid belt is stable enough - there are chances of changes of orbit there long term but nothing in the future few million years.

So the asteroid would be at most 10 kilometers or so in size. And actually, we have already found all the ten kilometer scale NEOs, it’s possible that there’s a ten kilometer scale object currently way beyond Jupiter - and that would be the largest one that we could be hit by on a decade timescale. The chance of this is tiny though, 99.99999% certain we won’t be hit by such a large asteroid in the next century (now that we know 100% of the NEO ten kilometer scale asteroids and none are headed our way).

Smaller asteroids are more common, so a 100 meter or so asteroid or smaller is the most likely scenario by far. Far more likely to predict one of those first, indeed many of those, before we get a kilometer scale asteroid.

For more details see the deflection section of my Giant Asteroid Headed Your Way? - How We Can Detect And Deflect Them and the other sections for things like the largest asteroid that could hit us.

It’s also available on kindle as

Giant Asteroid Is Headed Your Way? : How We Can Detect and Deflect Them (Amazon)